Counterexamples of Commutator Estimates in the Besov and the Triebel-Lizorkin Spaces Related to the Euler Equations

This paper deals with the Kato–Ponce-type commutator estimates in the Besov space $B^s_{p,q}(\mathbb{R}^n)$ and the Triebel–Lizorkin space $F^s_{p,q}(\mathbb{R}^n)$ related to the Euler equations describing the motion of perfect incompressible fluids. We investigate the relation between the optimal bound of the commutator estimates and the solvability of the Euler equations. In particular, we show that these commutator estimates fail in $B^s_{p,q}(\mathbb{R}^n)$ and $F^s_{p,q}(\mathbb{R}^n)$ with the critical differential order $s=n/p+1$ and various exponents p and q.